Properties of triangles in non-Euclidean geometries

Question

As we all know, the angles in all triangles in Euclidean geometry must add up to $180^\circ$. As some of us may know, this is not true in non-Euclidean geometries; for example, on the surface of a sphere, the angles in a triangle can be between $180^\circ$ and $540^\circ$, while on a hyperbolic paraboloid (a "saddle"), it can be anywhere between $0^\circ$ and $180^\circ$. This is just one property out of many that involve triangles in non-Euclidean geometries.

A Google search on this topic brought me too broad a range of references and sources. I understand that this has been studied on and worked on a lot. What other properties of triangles exist that are contrary to normal Euclidean geometry?

Answer 1

One neat example is that when considering triangles that lie on a sphere, similar triangles are congruent. That is, it is enough to know the angles of two triangles are the same to determine they are congruent because there is an absolute reference of magnitude, the radius of the sphere you consider.

Answer 2

In hyperbolic geometry there exists a "horizon", and size of similar triangles changes depending on how near it is to it, or its position.

A right angled triangle in Euclidean geometry $ c^2= a^2+b^2$.

In non- Euclidean hyperbolic and elliptic geometries

$$ \cosh c = \cosh a \cdot \cosh b , \cos c = \cos a \cdot \cos b $$

respectively for a unit radius of sphere or pseudosphere. Also

$$ \cos \frac{c}{R} = \cos \frac{a}{R} \cdot \cos \frac{b}{R} $$

and

$$ \cosh \frac{c}{R} = \cosh \frac{a}{R} \cdot \cosh \frac{b}{R} $$

for a sphere radius $R$ or pseudo-radius $R$.

Answer 3

On a sphere, every triangle may be associated with a dual triangle in which each vertex of either triangle is 90° of arc away from one side of the second triangle. Each side of either triangle is supplementary to the angle it faces in the second triangle; thus if you have a triangle with three arcs measuring 108° apiece, there must be a dual triangle with three angles measuring 72° apiece.

The existence of these dual triangles implies that any identity you have with spherical triangles may be replaced with one where each angle is replaced by the supplement of the opposite side and vice versa, which is equivalent to applying the identity to the dual triangle. For instance, if you accept that the arcs of a spherical triangle have less angular measure than thise of the small circle containing it, you have

$a+b+c<360°.$

The dual of this law, bounding the sum of the angles, is then

$(180°-A)+(180°-B)+(180°-C)<360°$

$\implies A+B+C>180°.$

Similarly, there are not one but two Laws of Cosines because one is the dual of the other:

$\cos c=\cos a\cos b +\sin a\sin b\cos C$

$\cos(180°-C)=\cos(180°-A)\cos(180°-B) +\sin(180°-A)\sin(180°-B)\cos(180°-c)$

$\implies\cos C= -\cos A\cos B+\sin A\sin B\cos c.$

Exercise for the reader: prove that the Spherical Law of Sines

$\dfrac{\sin A}{\sin a}=\dfrac{\sin B}{\sin b}=\dfrac{\sin C}{\sin c}$

is self-dual.